The problem of exciton migration and trapping on a linear polymer is treated as a random walk on a one-dimensional lattice. The average number of steps required for a walker to be trapped is calculated when the probability of stepping to adjacent lattice sites is not symmetrical, and is found to be less than that calculated for a symmetrical walk. An asymmetrical stepping probability is shown to result from the thermal vibrations of the lattice. The magnitude of this effect on the exciton lifetime is estimated and found to be significant.